Editing Harmonic function (section)

==Generalizations==

===Weakly harmonic function===
A function (or, more generally, a [[distribution (mathematics)|distribution]]) is [[weakly harmonic]] if it satisfies Laplace's equation
<math display="block">\Delta f = 0\,</math>
in a [[weak derivative|weak]] sense (or, equivalently, in the sense of distributions).  A weakly harmonic function coincides almost everywhere with a strongly harmonic function, and is in particular smooth.  A weakly harmonic distribution is precisely the distribution associated to a strongly harmonic function, and so also is smooth.  This is [[Weyl's lemma (Laplace equation)|Weyl's lemma]].

There are other [[weak formulation]]s of Laplace's equation that are often useful.  One of which is [[Dirichlet's principle]], representing harmonic functions in the [[Sobolev space]] {{math|''H''<sup>1</sup>(Ω)}} as the minimizers of the [[Dirichlet energy]] integral
<math display="block">J(u):=\int_\Omega |\nabla u|^2\, dx</math>
with respect to local variations, that is, all functions <math>u\in H^1(\Omega)</math> such that <math>J(u) \leq J(u+v)</math> holds for all <math>v\in C^\infty_c(\Omega),</math> or equivalently, for all <math>v\in H^1_0(\Omega).</math>

===Harmonic functions on manifolds===
Harmonic functions can be defined on an arbitrary [[Riemannian manifold]], using the [[Laplace–Beltrami operator]] {{math|Δ}}. In this context, a function is called ''harmonic'' if 
<math display="block">\ \Delta f = 0.</math>
Many of the properties of harmonic functions on domains in Euclidean space carry over to this more general setting, including the mean value theorem (over [[geodesic]] balls), the maximum principle, and the Harnack inequality.  With the exception of the mean value theorem, these are easy consequences of the corresponding results for general linear [[elliptic partial differential equation]]s of the second order.

===Subharmonic functions===
A {{math|''C''<sup>2</sup>}} function that satisfies {{math|Δ''f'' ≥ 0}} is called subharmonic.  This condition guarantees that the maximum principle will hold, although other properties of harmonic functions may fail.  More generally, a function is subharmonic if and only if, in the interior of any ball in its domain, its graph lies below that of the harmonic function interpolating its boundary values on the ball.

===Harmonic forms===
One generalization of the study of harmonic functions is the study of [[harmonic form]]s on [[Riemannian manifold]]s, and it is related to the study of [[cohomology]]. Also, it is possible to define harmonic vector-valued functions, or harmonic maps of two Riemannian manifolds, which are critical points of a generalized Dirichlet energy functional (this includes harmonic functions as a special case, a result known as  [[Dirichlet principle]]). This kind of harmonic map appears in the theory of minimal surfaces. For example, a curve, that is, a map from an interval in {{tmath|\R}} to a Riemannian manifold, is a harmonic map if and only if it is a [[geodesic]].

===Harmonic maps between manifolds===
{{main|Harmonic map}}
If {{mvar|M}} and {{mvar|N}} are two Riemannian manifolds, then a harmonic map <math>u: M \to N</math> is defined to be a critical point of the Dirichlet energy
<math display="block">D[u] = \frac{1}{2} \int_M \left\|du\right\|^2 \, d\operatorname{Vol}</math>
in which <math>du: TM \to TN </math> is the differential of {{mvar|u}}, and the norm is that induced by the metric on {{mvar|M}} and that on {{mvar|N}} on the tensor product bundle <math>T^\ast M \otimes u^{-1} TN.</math>

Important special cases of harmonic maps between manifolds include [[minimal surface]]s, which are precisely the harmonic immersions of a surface into three-dimensional Euclidean space.  More generally, minimal submanifolds are harmonic immersions of one manifold in another.  [[Harmonic coordinates]] are a harmonic [[diffeomorphism]] from a manifold to an open subset of a Euclidean space of the same dimension.